Unitary transformation

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Summary

In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.

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Operator preserving the Hilbert space structure
Unitary transformations are the operators that preserve the inner product in a Hilbert space, corresponding to time evolution in quantum mechanics.
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corresponds to rotations on the Bloch sphere
Any single-qubit unitary operation preserves the length of the Bloch vector and corresponds to a rotation of the sphere.
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describes the evolution of qudit states
The evolution of a qudit in a closed quantum system is governed by unitary transformations, which preserve the inner product of the state vectors.
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Facts